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Simplifying (4x + -4)(4x + 2) = 1 Reorder the terms: (-4 + 4x)(4x + 2) = 1 Reorder the terms: (-4 + 4x)(2 + 4x) = 1 Multiply (-4 + 4x) * (2 + 4x) (-4(2 + 4x) + 4x * (2 + 4x)) = 1 ((2 * -4 + 4x * -4) + 4x * (2 + 4x)) = 1 ((-8 + -16x) + 4x * (2 + 4x)) = 1 (-8 + -16x + (2 * 4x + 4x * 4x)) = 1 (-8 + -16x + (8x + 16x2)) = 1 Combine like terms: -16x + 8x = -8x (-8 + -8x + 16x2) = 1 Solving -8 + -8x + 16x2 = 1 Solving for variable 'x'. Reorder the terms: -8 + -1 + -8x + 16x2 = 1 + -1 Combine like terms: -8 + -1 = -9 -9 + -8x + 16x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -9 + -8x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.5625 + -0.5x + x2 = 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + -0.5x + 0.5625 + x2 = 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + -0.5x + x2 = 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + -0.5x + x2 = 0 + 0.5625 -0.5x + x2 = 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 -0.5x + x2 = 0.5625 The x term is -0.5x. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5x + 0.0625 + x2 = 0.5625 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 0.5625 + 0.0625 Combine like terms: 0.5625 + 0.0625 = 0.625 0.0625 + -0.5x + x2 = 0.625 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 0.625 Calculate the square root of the right side: 0.790569415 Break this problem into two subproblems by setting (x + -0.25) equal to 0.790569415 and -0.790569415.Subproblem 1
x + -0.25 = 0.790569415 Simplifying x + -0.25 = 0.790569415 Reorder the terms: -0.25 + x = 0.790569415 Solving -0.25 + x = 0.790569415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 0.790569415 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 0.790569415 + 0.25 x = 0.790569415 + 0.25 Combine like terms: 0.790569415 + 0.25 = 1.040569415 x = 1.040569415 Simplifying x = 1.040569415Subproblem 2
x + -0.25 = -0.790569415 Simplifying x + -0.25 = -0.790569415 Reorder the terms: -0.25 + x = -0.790569415 Solving -0.25 + x = -0.790569415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -0.790569415 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -0.790569415 + 0.25 x = -0.790569415 + 0.25 Combine like terms: -0.790569415 + 0.25 = -0.540569415 x = -0.540569415 Simplifying x = -0.540569415Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.040569415, -0.540569415}
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